Chapter 10

10.8 Evans Root Locus Construction Rule #6: Angles of Departures/Arrivals at Complex Poles/Zeros

This Rule deals with angles of departure (arrival) from/to complex poles (zeros).

 

Rule 6: If [latex]G(s)[/latex] has a pole p of multiplicity r, then r branches of the root locus depart from p. The angle of departure of these root loci from p are described by this equation:

[latex]\theta_{dep}[/latex]= [latex]\frac{ \angle [G(s)(s-p)^r]_{s=p}+(2k+1) \cdot 180^{\circ}}{r}[/latex]  [latex]k=0,1,2,...,r-1[/latex]                                                                                                                                                                                 Equation 10-13

Similarly, if [latex]G(s)[/latex] has a zero z of multiplicity r, then r branches of the root locus arrive at z. The angle of arrival of these root loci to z are described by this equation:

[latex]\theta_{arr} = \frac{-\angle[\frac{G(s)}{(s-z)^r}]_{s=z}+(2k+1) \cdot 180^{\circ}}{r} k=0,1,2,...,r-1[/latex]

Equation 10-14

 

See examples in the next section for illustration of this rule.

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